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Say I have annual observations of the temperatures at the North Pole and South Pole for many years. I want to build a model that given the South Pole temperature for the current year and all prior history can predict the North Pole temperature for the current year:

$$\widehat{T^{(North)}_i} = f \left(\left\{T^{(South)}_j\right\}_{j=1..i}, \left\{T^{(North)}_j\right\}_{j=1..i-1}\right)$$

What's an appropriate model here given that there will be significant serial correlation in the data? (from global and local climate/weather trends)

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The approach is called a Transfer Function or Dynamic Regression. The objective is to identify the relationship/model parameters and to incorporate any ARIMA structure reflecting omitted stochastic series and identifiable intervention series reflecting omitted deterministic series via Intervention Detection. Care must be taken to validate/test for/accommodate any evidence based parameter changes over time and any evidence based non-constancy in the error process over time. Model identification for the causal structure start with single pre-whitening procedures to form the Impulse Response Weights leading to the tentative form of the transfer between the inputs and the output. See http://www.autobox.com/cms/index.php/afs-university/intro-to-forecasting/doc_download/26-seminar-slide-show starting at slide 57 for some material relative to this.

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